Abstract

Let us consider groups G1=Zk∗(Zm∗Zn), G2=Zk×(Zm∗Zn), G3=Zk∗(Zm×Zn), G4=(Zk∗Zl)∗(Zm∗Zn) and G5=(Zk∗Zl)×(Zm∗Zn), where k,l,m,n≥2. In this paper, by defining a new graph Γ(Gi) based on the Gröbner-Shirshov bases over groups Gi, where 1≤i≤5, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of Γ(Gi). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in such fields as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. In addition, the Gröbner-Shirshov basis and the presentations of algebraic structures contain a mixture of algebra, topology and geometry within the purposes of this journal.

MSC:05C25, 13P10, 20M05, 20E06, 26C10.

Keywords

graphsGröbner-Shirshov basesgroup presentation

1 Introduction and preliminaries

In [1, 2], the authors have recently developed a new approach between algebra (in the meaning of groups and monoids) and analysis (in the meaning of generating functions). In a similar manner, in this paper, we make a connection between graph theory and Gröbner-Shirshov bases. In the literature, there are no works related to the idea of associating a graph with the Gröbner-Shirshov basis of a group. So, we believe that this paper will be the first work in that direction. As we depicted in the abstract of this paper, while graph theoretical studies actually consist of some fixed point techniques, so they have been applied in different branches of science such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), Gröbner-Shirshov bases and algebraic presentations contain a mixture of algebra, topology and geometry within the purposes of this journal.

In detail, in this paper, we investigate the interplay between the group-theoretic property of a group G and the graph-theoretic properties of Γ(G) which is associated with G. By group-theoretic property, while we deal with the Gröbner-Shirshov basis of a given group, by graph-theoretic property, we are interested in the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of the corresponding graph of group. In the literature, there are some important graph varieties and works that are related to algebraic and topological structures, namely, Cayley graphs [3–5] and zero-divisor graphs [6–8]. But the graph constructed in here is different from the previous studies and is also interesting in terms of using Gröbner-Shirshov basis theory during the construction of the vertex and edge sets. So, this kind of graph provides not only the classification of algebras (groups, semigroups), but also solving the problems of normal forms of elements, word problem, rewriting system, embedding theorems, extensions of groups and algebras, growth function, Hilbert series, etc. As is well known, these problems are really important in fixed point results since they have a direct connection to nature sciences.

Throughout this paper, for k,l,m,n≥2, we consider special groups G1=Zk∗(Zm∗Zn), G2=Zk×(Zm∗Zn), G3=Zk∗(Zm×Zn), G4=(Zk∗Zl)∗(Zm∗Zn) and G5=(Zk∗Zl)×(Zm∗Zn) associated with the presentations

respectively. By recalling the fundamentals of the Gröbner-Shirshov (GS) basis and then obtaining the GS-basis of each above group Gi, in Section 1.1, we define new simple, undirected graphs Γ(Gi) associated with GS-bases of these groups. Then in Section 2, we compute the diameter, maximum and minimum degrees, girth, chromatic, clique and domination numbers, degree sequence and finally irregularity index of graphs Γ(Gi) for each 1≤i≤5.

Remark 1 The reason for us to present our results on these above parameters actually comes from their equality status. In other words, each result will be a good example for certain equalities over graph theoretical theorems.

(I) Preliminaries for graph theory. We first recall that for any simple graph G, the distance (length of the shortest path) between two vertices u, v of G is denoted by dG(u,v). If no such path exists, we set d(x,y):=∞. Actually, the diameter of G is defined by

diam(G)=max{dG(x,y):x and y are vertices of G}.

The degree degG(v) of a vertex v of G is the number of vertices adjacent to v. Among all degrees, the maximum degreeΔ(G) (or the minimum degreeδ(G)) of G is the number of the largest (or the smallest) degrees in G ([9]).

It is known that the girth of a simple graph G, gr(G) is the length of the shortest cycle contained in G. However, if G does not contain any cycle, then the girth of it is assumed to be infinity.

Basically the coloring of G is an assignment of colors (elements of some set) to the vertices of G, one color to each vertex, so that adjacent vertices are assigned distinct colors. If n colors are used, then the coloring is referred to as n-coloring. If there exists an n-coloring of G, then G is called n-colorable. The minimum number n for which G is n-colorable is called a chromatic number of G and is denoted by χ(G). There exists another graph parameter, namely the clique of a graph G. In fact, depending on the vertices, each of the maximal complete subgraphs of G is called a clique. Moreover, the largest number of vertices in any clique of G is called the clique number and denoted by ω(G). In general, χ(G)≥ω(G) for any graph G [9] and it is worth studying the cases that imply equality.

A subset D of the vertex set V(G) of a graph is called a dominating set if every vertex V(G)∖D is joined to at least one vertex of D by an edge. Additionally, the domination numberγ(G) is the number of vertices in the smallest dominating set for G [9].

There also exists the term degree sequence, denoted by DS(G), which is a sequence of degrees of vertices of G. In [10], a new parameter for graphs has been recently defined, namely the irregularity index of G denoted by t(G), which is the number of distinct terms in the set DS(G).

(II) Preliminaries for the Gröbner-Shirshov basis. Since the main body of the paper is built up by considering the Gröbner-Shirshov (GS) basis, it is worth presenting some historical background about it as in the following.

The Gröbner basis theory for commutative algebras, which provides a solution to the reduction problem for commutative algebras, was introduced by Buchberger [11]. In [12], Bergman generalized the Gröbner basis theory to associative algebras by proving the ‘diamond lemma.’ On the other hand, the parallel theory of Gröbner bases was developed for Lie algebras by Shirshov [13]. The key ingredient of the theory is the so-called composition lemma which characterizes the leading terms of elements in the given ideal. In [14], Bokut noticed that Shirshov’s method works for associative algebras as well. Thus, for this reason, Shirshov’s theory for Lie algebras and their universal enveloping algebras is called the Gröbner-Shirshov basis theory. There are some specific studies on this subject related to some algebraic structures (see, for instance, [15–17]). We may finally refer to the papers [18–24] for some other recent studies of Gröbner-Shirshov bases. In the following, we give some fundamental facts about this important subject.

Let K be a field, and let K〈X〉 be the free associative algebra over K generated by X. Denote by X∗ the free monoid generated by X, where the empty word is the identity denoted by 1. For a word w∈X∗, we denote the length of w by |w|. Suppose that X∗ is a well-ordered set. Then every nonzero polynomial f∈K〈X〉 has the leading word f¯. If the coefficient of f¯ in f is equal to one, then f is called monic.

Let f and g be two monic polynomials in K〈X〉. We then have two compositions as follows:

If w is a word such that w=f¯b=ag¯ for some a,b∈X∗ with |f¯|+|g¯|>|w|, then the polynomial (f,g)w=fb−ag is called the intersection composition of f and g with respect to w. The word w is called an ambiguity of intersection.

If w=f¯=ag¯b for some a,b∈X∗, then the polynomial (f,g)w=f−agb is called the inclusion composition of f and g with respect to w. The word w is called an ambiguity of inclusion.

If g is monic, f¯=ag¯b and α is the coefficient of the leading term f¯, then transformation f↦f−αagb is called elimination (ELW) of the leading word of g in f.

Let S⊆K〈X〉 with each s∈S be monic. Then the composition (f,g)w is called trivial modulo (S,w) if (f,g)w=∑αiaisibi, where each αi∈K, ai,bi∈X∗, si∈S and aisi¯bi<w. If this is the case, then we write (f,g)w≡0mod(S,w). We call the set S endowed with the well-ordering < a Gröbner-Shirshov basis for K〈X∣S〉 if any composition (f,g)w of polynomials in S is trivial modulo S and corresponding w.

The following lemma was proved by Shirshov [13] for free Lie algebras with deg-lex ordering (see also [25]). In 1976, Bokut [14] specialized Shirshov’s approach to associative algebras (see also [12]). Meanwhile, for commutative polynomials, this lemma is known as Buchberger’s theorem (see [11, 26]).

Lemma 1 (Composition-diamond lemma)

LetKbe a field,

A=K〈X∣S〉=K〈X〉/Id(S)

and let < be a monomial ordering onX∗, whereId(S)is the ideal ofK〈X〉generated byS. Then the following statements are equivalent:

1

Sis a Gröbner-Shirshov basis.

2

f∈Id(S)⇒f¯=as¯bfor somes∈Sanda,b∈X∗.

3

Irr(S)={u∈X∗∣u≠as¯b,s∈S,a,b∈X∗}is a basis for the algebraA=K〈X∣S〉.

If a subset S of K〈X〉 is not a Gröbner-Shirshov basis, then we can add to S all nontrivial compositions of polynomials of S, and by continuing this process (maybe infinitely), we eventually obtain a Gröbner-Shirshov basis Scomp. Such a process is called the Shirshov algorithm.

1.1 A new graph based on GS-bases

In the following, for 1≤i≤5, by taking into account each group Gi presented by PGi=〈Xi;Ri〉, as in (1), we define a undirected graph Γ(Gi)=(Vi,Ei) and all results will be constructed on it.

The vertexVi and edgeEi={(vp,vq)} sets consist of

generators of Gi,

leading elements of polynomials in the GS basis of Gi,

ambiguities of intersection or inclusion in the GS basis of Gi,

and

vp and vq form an ambiguity with each other,

∃vr∈Xi∗ such that vr=xvp or vr=vqy for some x,y∈Xi,

vp is reducible to vq,

respectively.

Since the Gröbner-Shirshov basis plays an important role in the definition of this new graph, let us define these bases for each of the groups Gi where 1≤i≤5. To do that, let us assume an ordering among the generators of Gi (1≤i≤3) as x>a>b and the generators of G4 and G5 as x>y>a>b.

Now, let us first consider PG1. Since we have no intersection or inclusion compositions, the Gröbner-Shirshov basis of G1 is S1={xk−1,am−1,bn−1}.

For PG2, we have the ambiguities of intersection as xka, xam, xkb, xbn. Since these are trivial to modulo R2, the Gröbner-Shirshov basis of G2 is S2={xk−1,am−1,bn−1,xa−ax,xb−bx}.

For PG3, we have the ambiguities of intersection as amb and abn. Since these are trivial to modulo R3, the Gröbner-Shirshov basis of G3 is S3={xk−1,am−1,bn−1,ab−ba}.

For PG4, since we have no intersection or inclusion compositions, the Gröbner-Shirshov basis of G4 is S4={xk−1,yl−1,am−1,bn−1}.

Finally, for PG5, we have the ambiguities of intersection as

xka,xam,xkb,xbn,yla,yam,ylb,ybn.

Since these are trivial to modulo R5, the Gröbner-Shirshov basis of G5 is S5={xk−1,yl−1,am−1,bn−1,xa−ax,xb−bx,ya−ay,yb−by}.

2.1 Case 1: the graph Γ(G1), where G1=Zk∗(Zm∗Zn)

If we consider the graph of the group G1, then we have a subgraph of Figure 1(a) with vertices v1=am, v2=a, v4=xk, v5=x, v9=b and v10=bn. In this graph the edge set depends on the orders of factor groups of G1. If we take k,m,n=2, then by the edge definition, we have the edges α1, α4, α15 in this subgraph of Figure 1(a). In the case k,m,n>2, we do not have any edges. In the remaining case, i.e., one or two orders of factor groups of G1 are equal to two, we have one or two edges among α1, α4 and α15.

Figure 1

Modals of the new graph. (a) The general graph Γ(Gi) based on the Grobner-Shirshov basis. (b) The graph Γ(G), where G=Z3×(Z3∗Z3) as defined in Example 1.

If we reconsider that the graph in Figure 1(a) depends on the group G1 with above facts, then the picture will be shown as an unconnected graph which is not related to the numbers k, m and n. Thus the first result is the following.

Theorem 1diam(Γ(G1))=∞.

Theorem 2The maximum and minimum degrees of the graphΓ(G1)are

Δ(Γ(G1))={1;at least one ofk,m,n=2,0;k,m,n>2

and

δ(Γ(G1))={1;k,m,n=2,0;at most two ofk,m,n=2ork,m,n>2,

respectively.

Proof If we take that at least one of k, m, n is two, then we have at least one of the edges α1, α4, α15. Thus we get Δ(Γ(G1))=1. If we consider k=m=n=2, then since we have edges α1, α4, α15, this gives us that all vertices in the graph Γ(G1) have degree one. So, Δ(Γ(G1))=δ(Γ(G1))=1. Now we consider the case k,m,n>2. In this case, since we have no edges in the graph Γ(G1), we obtain Δ(Γ(G1))=δ(Γ(G1))=0. If we take that at most two of k, m, n are equal to two, then we get four vertices having degree one and two vertices having degree zero. Therefore, in this case, δ(Γ(G1))=0. □

Theorem 3For anyk,m,n≠1, the girth of the graphΓ(G1)is equal to infinity.

Proof Since we just have the edges α1, α4 and α15 depending on the numbers k, m, n, we do not have any cycle in the graph Γ(G1). So, gr(Γ(G1))=∞. □

Theorem 4The chromatic number ofΓ(G1)is equal to

χ(Γ(G1))={2;at least one ofk,m,n=2,1;k,m,n>2.

Proof If we take that at least one of k, m, n is two, then we have at least one of the edges α1, α4, α15. Thus we use two different colors since there exist neighbor vertices. By the edge definition of Γ(G1), we do not have any edges between generators and elements of three factor groups of G1. Thus we obtain χ(Γ(G1))=2. If we consider k,m,n>2, then since we do not have any edges in the graph, we can label all vertices with the same color. Therefore χ(Γ(G1))=1. □

Theorem 5The clique number ofΓ(G1)is equal to

ω(Γ(G1))={2;at least one ofk,m,n=2,1;k,m,n>2.

Proof The proof of this theorem is similar to the proof of Theorem 4. If we take that at least one of k, m, n is 2, then we have at least one of the edges α1, α4, α15, i.e., we have a disconnected graph which has at least three complete subgraphs. Since these complete subgraphs have two vertices, we get ω(Γ(G1))=2. If we consider k,m,n>2, then since we do not have any edges in the graph, the number of vertices in maximal clique is one. □

Theorem 6The domination number ofΓ(G1)is equal to infinity.

Proof For all cases of k, m, n, since the graph Γ(G1) is disconnected, we get γ(Γ(G1))=∞. □

Theorem 7The degree sequence and irregularity index ofΓ(G1)are given by

Proof By the graph Γ(G1), if k, m, n are equal to two and greater than two, then the degrees of the vertices are one and zero, respectively. But if at least one of k, m, n is equal to two, then some vertices have degree one and some of them have degree zero. Hence, by the definition of a degree sequence, we clearly obtain the set DS(Γ(G1)), as depicted. Nevertheless, it is easily seen that the irregularity index t(Γ(G1))=1 and 2, as required. □

2.2 Case 2: the graph Γ(G2), where G2=Zk×(Zm∗Zn)

If we consider the graph of the group G2, then we have a subgraph of Figure 1(a) with vertices v1=am, v2=a, v3=xka, v4=xk, v5=x, v6=xam, v7=xa, v8=xkb, v9=b, v10=bn, v11=xbn and v12=xb. In this graph the edge set depends on the orders of factor groups of G2. If we take k,m,n=2 then, by the edge definition, we have the edges αj, 1≤j≤23. In the case k,m,n>2, we do not have edges α1, α4, α9, α12, α15, α20 and α23 in Γ(G2).

Theorem 8The diameter of the graphΓ(G2)is equal to four.

Proof By considering the graph of the group G2, we say that the diameter of the graph Γ(G2) does not depend on the numbers k, m, n. For any k, m, n, in the graph Γ(G2) the vertices v7=xa and v12=xb are adjacent to vertices v1, v2, v3, v4, v5, v6 and v4, v5, v8, v9, v10, v11, respectively. If we connect any two vertices, except v4 and v5, via the shortest path, we need to pass through the vertices v7 and v12. For instance, we need the edges α7, α11, α18 and α22 to connect two vertices v1=a and v10=b. This gives us diam(Γ(G2))=4. □

Theorem 9The maximum and minimum degrees of the graphΓ(G2)are

Δ(Γ(G2))={6;k,m,n=2,4;k,m,n>2

and

δ(Γ(G2))={3;k,m,n=2,2;k,m,n>2,

respectively.

Proof For k,m,n=2, in the graph of G2 the vertices v7=xa and v12=xb are adjacent to vertices v1, v2, v3, v4, v5, v6 and v4, v5, v8, v9, v10, v11, respectively. Since these vertices have the largest degrees in Γ(G2), we get Δ(Γ(G2))=6. The other vertices v1, v2, v3, v6, v8, v9, v10 and v12 have degree three and the remaining vertices v4 and v5 have degree five. So, the minimum degree of the graph Γ(G2) is δ(Γ(G2))=3. Now we take k,m,n>2. In this case, we do not have edges α1, α4, α9, α12, α15, α20 and α23. Thus the vertices v4, v5, v7, v12 have degree four and the remaining vertices have degree two. So, Δ(Γ(G2))=4 and δ(Γ(G2))=2. □

Theorem 10The girth of the graphΓ(G2)is equal to

gr(Γ(G2))={3;k,m,n=2,4;k,m,n>2.

Proof By the considering the graph of the group G2, we have twelve triangles and five squares for k=m=n=2 and k,m,n>2, respectively. By the definition of girth, this gives us the required result. □

Theorem 11The chromatic number of the graphΓ(G2)is equal to

χ(Γ(G2))={3;at least one ofk,m,n=2,2;k,m,n>2.

Proof If the graph Γ(G2) has one of the following forms: Z2×(Zm∗Zn), Zk×(Z2∗Zn), Zk×(Zm∗Z2), Z2×(Z2∗Zn), Z2×(Zm∗Z2), Zk×(Z2∗Z2) or Z2×(Z2∗Z2), then we have similar neighbors for the graphs of each group. So, we can label the vertices with three different colors. If k,m,n≠2, then in the graph of G2 each vertex has two or four neighbors. In this graph, since the opposite vertices, which have an edge between them, can be labeled with the same color, we have two different colors. Hence χ(Γ(G2))=2. □

Theorem 12The domination number of the graphΓ(G2)is

γ(Γ(G2))={2;k,m,n=2,4;k,m,n>2.

Proof Firstly, we consider the case k=m=n=2. Since the vertices v7=xa and v12=xb are connected with all other vertices in the graph of G2, we can take the dominating set as {v7,v12}. Thus γ(Γ(G2))=2. In the case k,m,n≠2, since the number of edges is decreasing, the number of connected edges is decreasing as well. In this case, let us choose the dominating set as {v4,v5,v7,v12}. Every vertex, except the vertices in the dominating set, is joined to at least one vertex of this dominating set by an edge. Therefore we have γ(Γ(G2))=4. □

Theorem 13The clique number ofΓ(G2)is equal to

ω(Γ(G2))={3;at least one ofk,m,n=2,2;k,m,n>2.

Proof In the graph Γ(G2), for k=m=n=2, we have twelve complete subgraphs. These are obtained by the vertices v1−v2−v7, v2−v3−v7, v3−v4−v7, v4−v5−v7, v5−v6−v7, v4−v5−v12, v4−v8−v12, v8−v9−v12, v9−v10−v12, v10−v11−v12 and v5−v11−v12. Hence ω(Γ(G2))=3. If k,m,n≠2, then we can find the smallest complete subgraphs as edges obtained by any two vertices in the graph Γ(G2). So, ω(Γ(G2))=2. □

Theorem 14The degree sequence and irregularity index ofΓ(G2)are given by

and x>a>b, the graph Γ(G) as drawn in Figure 1(b), with the vertex set

V(Γ(G))={x,a,b,x3,a3,b3,x3a,x3b,xa3,xb3,xa,xb}.

By the result of Theorems 8 and 14, we have diam(Γ(G))=4, Δ(Γ(G))=4, δ(Γ(G))=2, gr(Γ(G))=4, χ(Γ(G))=2, γ(Γ(G))=4, ω(Γ(G))=2, DS(Γ(G))=(2,2,2,4,4,2,4,2,2,2,2,4) and t(Γ(G))=2.

2.3 Case 3: the graph Γ(G3), where G3=Zk∗(Zm×Zn)

If we consider the graph of the group G3, then we have a subgraph of Figure 1(a) with vertices v1=am, v2=a, v3=abn, v4=bn, v5=b, v6=amb, v7=ab, v9=x and v10=xk. If we take k,m,n=2, then by the edge definition, we have the edges αj, 1≤j≤12 and α15 in this subgraph of Figure 1(a). In the case k,m,n>2, we do not have edges α1, α4, α9, α12, α15 in the graph Γ(G3).

Theorem 15The maximum and minimum degrees of the graphΓ(G3)are

Δ(Γ(G3))={6;k,m,n=2,4;k,m,n>2

and

δ(Γ(G3))={1;k,m,n=2,0;k,m,n>2,

respectively.

Proof Let us consider the graph Γ(G3) and take k,m,n=2. In this case, the vertex v7 has the maximum degree six and the vertices v9 and v10 have the minimum degree one. But if we take k,m,n>2, then since there do not exist the edges α1, α4, α9, α12 and α15 in the graph Γ(G3), we obtain the maximum degree four by the vertex v7 and the minimum degree zero by the vertices v9 and v10. □

Theorem 16The girth of the graphΓ(G3)is

gr(Γ(G3))={3;k,m,n=2,4;k,m,n>2.

Proof Firstly, we take account of the case k=m=n=2. In this case, we have six triangles which have the edges α1−α7−α8, α2−α8−α9, α3−α9−α10, α4−α10−α11, α5−α11−α12, α6−α7−α12 in the graph Γ(G3). Thus gr(Γ(G3))=3. Now we consider the case k,m,n>2. In this case, since we do not have the edges α1, α4, α9, α12 and α15, we have two squares which have the edges α1−α6−α5−α7 and α2−α3−α4−α7 in the graph Γ(G3). Therefore gr(Γ(G3))=4. □

Theorem 17The chromatic number of the graphΓ(G3)is

χ(Γ(G3))={3;k,m,n=2,2;k,m,n>2.

Proof Let us take k=m=n=2. In the graph Γ(G3), since the vertex v7 is connected with all vertices except the vertices v9 and v10, this vertex must be labeled by a different color than other vertices. In addition, since other vertices are connected with each other doubly, they can be labeled by two different colors. This gives us χ(Γ(G3))=3. In the case k,m,n>2, since we have two squares, as in the previous proof, in the graph Γ(G3), it is enough to label two adjacent vertices by different colors. Hence χ(Γ(G3))=2. □

Theorem 18The domination number of the graphΓ(G3)is equal to infinity.

Proof For all cases of k, m, n, since the graph Γ(G3) is disconnected, we get γ(Γ(G1))=∞. □

Theorem 19The clique number of the graphΓ(G3)is equal to

ω(Γ(G3))={3;k,m,n=2,2;k,m,n>2.

Proof For the case k=m=n=2, we have six maximal complete subgraphs of the graph Γ(G3) which are triangles as in the proof of Theorem 16. Thus the largest number of the vertices in any maximal complete subgraph is three. If we take k,m,n>2, then we get eight maximal complete subgraphs, namely α2, α3, α5, α6, α7, α8, α10 and α11, having two vertices. So, ω(Γ(G3))=2. □

Theorem 20The degree sequence and the irregularity index ofΓ(G3)are given by

DS(Γ(G3))={(3,3,3,3,3,3,6,1,1);k,m,n=2,(2,2,2,2,2,2,4,0,0);k,m,n>2

and

t(Γ(G3))=3;k,m,n≥2,

respectively.

Proof It is easily seen by the graph of the group G3. □

2.4 Case 4: the graph Γ(G4), where G4=(Zk∗Zl)∗(Zm∗Zn)

If we consider the graph of the group G4, then we get a subgraph Γ(G4) of the graph in Figure 1(a) with vertices v1=am, v2=a, v4=xk, v5=x, v9=b, v10=bn, v14=yl and v15=y. If we take k,l,m,n=2, then by Section 1.1, we obtain the edges α1, α4, α15 and α26 in Γ(G4). For the case k,l,m,n>2, we do not have any edges. On the other hand, since at most three orders of factor groups of G4 are equal to two, we have at most three edges between α1, α4, α15 and α26.

Since the proof of each condition of the next result is quite similar to the related results over the group G1=Zk∗(Zm∗Zn) in Case 1, we omit it.

Theorem 21Let us consider the groupG4=(Zk∗Zl)∗(Zm∗Zn)with its subgraphΓ(G4)as defined in the first paragraph of this case.

(i)

The maximum and minimum degrees of the graphΓ(G4)are

Δ(Γ(G4))={1;at least one ofk,m,n=2,0;k,m,n>2

and

δ(Γ(G4))={1;k,m,n=2,0;k,m,n>2,

respectively.

(ii)

For anyk, l, m, ndifferent from one by considering a subgraph of Figure 1(a), the girth of the graphΓ(G4)isgr(Γ(G4))=∞.

2.5 Case 5: the graph Γ(G5), where G5=(Zk∗Zl)×(Zm∗Zn)

Similarly as in Case 4, for the group G5, we obtain a subgraph Γ(G5) of the graph in Figure 1(a) having vertices v1=am, v2=a, v3=xka, v4=xk, v5=x, v6=xam, v7=xa, v8=xkb, v9=b, v10=bn, v11=xbn, v12=xb, v13=ylb, v14=yl, v15=y, v16=ybn, v17=yb, v18=yla, v19=v2=a, v20=v1=am, v21=yam and v22=ya. In this graph, the edge set depends on the orders of factor groups of G5. If we take k,l,m,n=2, then, by the adjacency definition in Section 1.1, we have the edges αj, 1≤j≤45 with α1=α37. For the case k,l,m,n>2, we do not have any edges α1, α4, α9, α12, α15, α20, α23, α26, α31, α34, α37=α1, α42, α45 in Γ(G5).

In the following result (Theorem 22 below), we again omit the proof of it as in Theorem 21 since it is quite similar to the related results over the group G2=Zk×(Zm∗Zn) in Case 2.

Theorem 22Let us consider the groupG5=(Zk∗Zl)×(Zm∗Zn)with its related graphΓ(G5)as defined in the first paragraph of Case 5.

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